The condition that the straight line lx + my = n may be a normal to the hyperbola $$b^2x^2 - a^2y^2 = a^2b^2$$ is given by

$$l^2/a^2 + m^2/b^2 = (a^2 - b^2)^2/n^2$$

$$a^2/l^2 + b^2/m^2 = (a^2 + b^2)^2/n^2$$

$$l^2/a^2 - m^2/b^2 = (a^2 + b^2)^2/n^2$$

$$a^2/l^2 - b^2/m^2 = (a^2 - b^2)^2/n^2$$

The vertices of a hyperbola are at (0, 0) and (10, 0) and one of its foci is at (18, 0). The equation of the hyperbola is

$$(x – 5)^2/25 – y^2/144 = 1$$

$$x^2/25 – (y – 5)^2/144 = 1$$

$$(x – 5)^2/25 – (y – 5)^2/144 = 1$$

If the straight line $$x \cos \alpha + y \sin \alpha = p$$ be a tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ , then

$$a^2 cos^2 \alpha + b^2 sin^2 \alpha = p^2$$

$$a^2 sin^2 \alpha + b^2 cos^2 \alpha = p^2$$

$$a^2 cos^2 \alpha + b^2 sin^2 \alpha = p^2$$

Given $$2b^2/a = 8$$ and $$3/\sqrt{5} = \sqrt{1 + b^2/a^2}$$ or $$4/5 = b^2/a^2$$ , find the required equation of the hyperbola.

The required equation of the hyperbola is $$x^2/25 - y^2/20 = 1$$ or $$4x^2 - 5y^2 = 100$$ .

The required equation of the hyperbola is $$x^2/20 - y^2/25 = 1$$ or $$5x^2 - 4y^2 = 100$$ .

The required equation of the hyperbola is $$x^2/16 - y^2/9 = 1$$ or $$9x^2 - 16y^2 = 100$$ .

The required equation of the hyperbola is $$x^2/9 - y^2/16 = 1$$ or $$16x^2 - 9y^2 = 100$$ .

19. (a) S(1,1), directrix is 2x + y = 1 and e = $$\sqrt{3}$$ . Now let the various point be (h, k), then accordingly $$\frac{\sqrt{(h-1)^2 + (k-1)^2}}{2h + k - 1}\cdot\frac{1}{\sqrt{5}} = \sqrt{3}$$ . On simplification, the required locus is _______.

$$7x^2 + 12xy - 2y^2 - 2x + 4y - 7 = 0$$

$$5x^2 + 8xy - 3y^2 - x + 2y - 5 = 0$$

$$6x^2 + 10xy - y^2 - 3x + 3y - 6 = 0$$

$$8x^2 + 14xy - 4y^2 - 4x + 6y - 8 = 0$$

23. (a) Equation of normal at any point (x1, y1) on hyperbola is,

$$a^2x1x - b^2y1y + (b^2 - a^2) = 0$$

$$a^2x1x + b^2y1y + (b^2 - a^2) = 0$$

Any normal to the hyperbola is given by the equation. Fill in the blank: The equation of any normal to the hyperbola is ________.

$$\frac{a x}{\sec \theta} - \frac{b y}{\tan \theta} = a^2 + b^2$$

$$\frac{a x}{\tan \theta} + \frac{b y}{\sec \theta} = a^2 - b^2$$

$$\frac{a x}{\cos \theta} - \frac{b y}{\sin \theta} = a^2 + b^2$$

$$\frac{a x}{\sin \theta} + \frac{b y}{\cos \theta} = a^2 - b^2$$

The formula for the eccentricity of a hyperbola is ________.

e = $$\sqrt{1 + \frac{b^2}{a^2}}$$

$$e = \sqrt{1 - \frac{b^2}{a^2}}$$

$$e = \sqrt{1 + \frac{a^2}{b^2}}$$

$$e = \sqrt{1 - \frac{a^2}{b^2}}$$

Let the hyperbola be $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ . Its conjugate will be $$\frac{x^2}{b^2} - \frac{y^2}{a^2} = 1$$ . The equation of the conjugate hyperbola is:

$$\frac{x^2}{b^2} - \frac{y^2}{a^2} = 1$$

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

$$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$

The formula for the eccentricity of the hyperbola is ________.

$$e = \sqrt{\frac{a^2 + b^2}{a^2}}$$

$$e = \sqrt{\frac{a^2 - b^2}{a^2}}$$

$$e = \sqrt{\frac{a^2 + b^2}{b^2}}$$

$$e = \sqrt{\frac{b^2 - a^2}{a^2}}$$

Given, equation of hyperbola is $$x^2 - 3y^2 = 2x + 8$$ . What is the conjugate of this hyperbola?

$$\frac{(x-1)^2}{9} + \frac{y^2}{3} = 1$$

$$\frac{(x-1)^2}{9} - \frac{y^2}{3} = 1$$

$$\frac{(x+1)^2}{9} - \frac{y^2}{3} = 1$$

$$\frac{(x+1)^2}{9} + \frac{y^2}{3} = 1$$

HYPERBOLA (EAMCET) EXERCISE

12th Grade

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100 Qs

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HYPERBOLA (EAMCET) EXERCISE

Quiz

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Mathematics

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12th Grade

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Practice Problem

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Hard

VASU PAGOTI

Used 5+ times

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100 questions

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MULTIPLE CHOICE QUESTION

45 sec • 1 pt

If the latus rectum of an hyperbola be 8 and eccentricity be 3/√5, then the equation of the hyperbola is

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

The length of the transverse axis of a hyperbola is 7 and it passes through the point (5, -2). The equation of the hyperbola is

None of these

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

If (0, ± 4) and (0, ± 2) be the foci and vertices of a hyperbola, then its equation is

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

The equation of the hyperbola whose directrix is x + 2y = 1, focus (2, 1) and eccentricity 2 will be

None of these

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

The equation of a hyperbola, whose foci are (5, 0) and (-5, 0) and the length of whose conjugate axis is 8, is

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is 16 and eccentricity is √2, is

MULTIPLE CHOICE QUESTION

45 sec • 1 pt

The equation of the hyperbola referred to the axis as axes of co-ordinate and whose distance between the foci is 16 and eccentricity is √2, is

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